In 1000, 1001, and 1003 the Khitan repeatedly attacked Song fortifications, and each time retreated without actually capturing them or making permanent territorial gains. Now to me, that sounds like a prime example of the "absolutely nomadic cavalry" being poor at besieging fortifications. The tribute was just a tired empire seeking peace, a goal which it succeeded at, and was not made because the Song were facing destruction otherwise: the Song actually won the defense of Chanyuan and this is why the Khitan were willing to stop raiding and sign a treaty.
That quote isn't implying that horse archers can't be effective (I've already said I agree they could be an effective strategy, so I don't know why you're trying to make it look like I'm arguing with you). Instead, I'm using horse archers failing in battle against horse archers to make an obvious point: even the most effective tactics can still fail due to other factors.
And yet, there are scores of tribes of horse archer tribes who never survived the Mongols. Because horse archery, like any other tactic, can be defeated even if it is effective. If you do consider a state's survival an example of efficacy of a tactic, then again just look at Hungary and Poland.
The (admittedly unsourced) number I saw was 600,000 for the Mongols. I can't find your source anywhere to see the context you've left out, as all copies of that Cambridge text are paywalled. At any rate even your source sounds quite numerically close, and it's not as simple as raw numbers when defection was so extremely common among the disaffected Song nobility, who in many cases surrendered entire armies/cities/fortresses and joined the Mongols without a fight due to dislike of the Song emperor. Quickly 100,000 Song men could become 100,000 Mongol men, which meant the Mongols could overcome their inability to besiege by using the traitor locals' siege expertise.
What about the First invasion then? During the Siege of Esztergom they had 30 siege engines. These reduced the wooden fortifications but failed against the stone castle. The Mongols then suffered heavy casualties from crossbowmen while trying to assault the castle. Because they suck at sieges. Also, why are we counting non-Mongols from areas nowhere near Mongolia as "Mongols" in the first place? If for large periods of time the Mongols were bad at sieges when they didn't have other groups available to help them, it's a very safe statement to say that Mongols were bad at siegeing stone fortifications.
This argument has only lasted this long because you refuse to address my two key points.
1: The invasions of Poland and Hungary were a prime before-and-after example of how effective crossbow/stone castle/heavy cavalry tactics could be against the Mongols.
2: Using the Song as an example of crossbowmen/fortifications not being effective at stopping Mongols is like using a DIY carpenter hitting their thumb as an example of a hammer not being effective at driving nails. The Song were an incredibly incompetent, inefficient bureaucracy which made multiple crucial mistakes, had no experienced generals, had an army mainly consisting of barely-trained peasant militia, and often had its forces defect without a fight, and their collapse was inevitable. They couldn't use the tools they had available to their proper effectiveness. Despite this,
their fortifications kept them alive for much longer than they would have lasted otherwise. The Mongols would have wiped them out in 5 years instead of 45. But instead the bungling Song were one of the most difficult cultures for the Mongols to conquer. Once you acknowledge that, the Song become a positive example and the argument is over. Which is why you keep skirting around it and pushing the argument towards tangential matters.
Bannerlord isn't meant to be all of Eurasia though. It represents only part of the Near East, Europe, and North Africa; a total area the Mongols only ever conquered about a sixth of. More importantly, Khuzaits aren't really Mongols! They only have partial Mongol inspiration, as Bannerlord's setting takes cues from 600-1000, well before the Mongols ever reached Europe.
Khuzaits are mainly based on Gokturks, Avars, Kipchaks, and Khazars. These groups never conquered anywhere near as much as they do ingame. They were actually wiped out, for the most part.
The Roman empire utilized cavalry but the bulk of its army's strength was in its footsoldiers, and this applies for plenty of other large empires throughout history.